Category Archives: Mathematics

Post navigation

There are many milestone books in the history of science. True to the definition of “milestone,” these works mark pivotal achievements in mankind’s effort to understand natural law and the world in which we live. Among the very top tier of milestone science books is the Discorsi, published in 1638 by Galileo Galilei. It is his last and most significant contribution to science and is often referred to as the first physics “textbook.”

The Discorsi derives its fame as the published repository of Galileo’s life-long efforts to decipher the concept of motion and the “law of fall,” the mathematical characterization of free-falling bodies under the influence of gravity. Attempts to determine the velocity and distance profiles of falling bodies as a function of elapsed time of fall had yielded only conjecture over the centuries since Aristotle himself pondered the question. As Aristotle observed over two thousand years ago, “In order to know the natural world, one must first understand motion,” and, until Galileo and Isaac Newton came along, what we “knew” about motion was indeed largely conjecture – much of it erroneous!
It was clear that the force of gravity was the prime-mover causing bodies of mass to fall toward the earth’s center, but until Isaac Newton in 1687 established how gravity worked, no one really understood the mathematical details of gravitational attraction between any two bodies of mass. When we weigh ourselves on a scale, we are, in fact, measuring the force of gravitational attraction between the earth as one mass and our bodies as the other. One’s weight on the moon is less than that on earth because of the moon’s smaller mass. In empty space, we are weightless. Here, then, is the great question posed by the elusive “law of fall” whose answer had eluded man for centuries – until Galileo solved the riddle:

What is the velocity attained by a free-falling body as a function of the elapsed time of fall? We know the velocity at the onset of fall: It is clearly zero at the instant of release. But what is its precise instantaneous velocity at each second of elapsed time thereafter? Furthermore, what is the distance of fall covered from the release point at each elapsed second of time?

A falling body precisely one second into free-fall is already traveling with an instantaneous velocity of 32.2 feet per second. Making such measurements without modern instrumentation and high-speed strobe photography would be virtually impossible. In the early seventeenth century when photography was unimaginable and there certainly were no stopwatches or even accurate clocks, Galileo had to find a way to slow down the motion of a free-falling body. He “diluted” the effect of gravity by repeatedly rolling a small ball from a standing start down a long grooved, inclined plane. He then measured the distances covered by the slowly accelerating ball over successive constant-time intervals of an arbitrary “musical beat.” It is said that he used for a “timing clock” the steady cadence of a hummed Italian march for his equal timing intervals. Using multiple trials and noting the precise position of the rolling ball along the track on successive numbers of downbeats of the steady cadence, he was able to deduce the “law of fall.” Here is a figure illustrating the essence of Galileo’s brilliant inclined plane experiment. The figure is from chapter four of my book on motion, The Elusive Notion of Motion: The Genius of Kepler, Galileo, Newton, and Einstein.

Here is the critical essence of what Galileo determined:

A body free-falling under gravity exhibits a constant acceleration value (near the earth’s surface). It remained for Galileo’s successors to determine the exact numerical value of that acceleration. Near the earth’s surface, a body of mass in free-fall attains an additional velocity of 32.2 feet per second for each additional second of elapsed time. Galileo’s determination that acceleration is constant in free-fall dictates two major conclusions:

-The velocity attained is proportional to the elapsed time of fall. Twice the elapsed time, twice the velocity, for example.

-The distance traveled is proportional to the square of the elapsed time of fall. Twice the elapsed time, four times the distance traveled.

This is the celebrated “law of fall.” There might be a tendency for the casual reader to shrug-off Galileo’s achievement as no big deal in light of modern scientific achievements, but it was a VERY big deal for the progress of physics. Galileo, along with Johannes Kepler and Kepler’s experimentally formulated three laws of planetary motion, paved the way for the truly great reformation in mathematical physics initiated by Isaac Newton in his masterwork book of 1687, the Principia, universally acclaimed as the greatest scientific book ever published.

Image: Pierre Barge & Associates Auctions

First-state presentation copy of Galileo’s Discorsi to the French Ambassador, Count Francois de Noailles who smuggled the manuscript from Florence, Italy, to Leiden, Holland, for publication in 1638. Auctioned in Paris for over $790,000 in April, 2017 by Pierre Barge & Associates, the book itself is dedicated to de Noailles.

Here is perhaps the finest copy extant of Galileo’s Discorsi E Dimostrazioni Matematiche intorno a due nuoue scienze, otherwise known as Discourses on Two New Sciences. This one-of-a-kind presentation copy from 1638 was given to a friend of Galileo’s who smuggled the final portion of Galileo’s manuscript out of Florence, Italy, into Leiden, Holland for publication by the famed Elzevir Press. Galileo was being held in virtual house arrest within his villa outside of Florence by mandate from the Catholic Church and its Inquisition. Galileo had been accused of suspicion of heresy by the Church for his previous 1632 book, the Dialogo, which the Church felt promoted the Copernican “world system” which featured a sun-centered solar system. This went against established scripture which suggested that the earth was at the center of everything, according to the Church. Galileo famously maintained that religion’s role on earth should be to show the way to heaven; it should be the role of science, not the Church, to explain the clockwork of the heavens. The Church did not agree.

As for the “two new sciences” introduced by Galileo’s Discorsi: The first treatise in the book deals with what would today be called “Strength of Materials” and “Reliability Engineering.” This constituted a pioneering effort by Galileo in a new field of endeavor. The second treatise in the book involves those categories of physics known as “Mechanics,” and “Kinematics.” The centerpiece of the book is, of course, Galileo’s findings on motion and the long-delayed, finally published documentation of the “law of fall.” Most of the work presented in that section was done by Galileo as early as 1604. Galileo was in poor health and almost blind from glaucoma in 1638; accordingly, publication of this, his most important scientific work was, for him, a very high priority made complicated by the censorship of the Church. Galileo died in 1642, the year Isaac Newton came into this world. The torch had been passed.

Much has been written about the growing disparity between the test scores of Chinese and American students – especially in science and math. Yesterday’s San Jose Mercury News carried a preview of a new book titled Little Soldiers: An American Boy, a Chinese School, and the Global Race to Achieve.

This is a subject near and dear to my heart, so I naturally checked Amazon for the book. I was pleased to find that the book became available that very day. Based on the impressive newspaper review of the book and the author’s obvious writing ability, I ordered a copy and look forward to reading it, soon.

Why am I so interested in the general subject of students and their education? For two reasons: First, I was fortunate to be the first in my entire family tree to attend and graduate from college – many years ago (B.S. Electrical Engineering, Stanford University, 1963). Second, my wife and two grown daughters all taught/teach school. Accordingly, I have a great appreciation of the benefits from a good education as well as the difficulties teachers, today, encounter in school classrooms.

What are those difficulties in American schools? The core of the problems centers on poor student attitudes toward school and learning and too much leeway given to students, their parents, and school administrators – at the expense of teachers, classroom discipline, and effective education. I offer a concrete example.

My oldest daughter teaches in grades 1-3 in the public schools. In each of her classes for the last three years, she has been saddled with a different and singularly difficult student, one who sapped much of her time and energy each day in class and after class. Each of these youngsters would, in past years, have been cited as special education students – students with significant learning/attention/ behavior disabilities. Today’s educational philosophies embrace the policies of “mainstreaming,” or “inclusion” whereby such challenged students are placed in regular classes as opposed to special education classes where small classes of special needs students are capably handled by trained special-ed teachers. The thinking behind this recent policy of inclusion? Immersion in a regular class will benefit the disadvantaged student by minimizing stigma while conditioning the other students’ understanding and empathy toward those with problems.

The reality? A regular classroom which accommodates a special needs student with significant learning/attention/behavior problems is often a nightmare for the teacher and a detriment to the learning environment for the other students. One such student my daughter has encountered continually disrupted the class with unprovoked behaviors such as screaming, throwing objects at the other students (and the teacher), kicking other students and sometimes bolting from the classroom. Heeding directives from the teacher seemed void of priority.

The moral of that story: One child who should not be in a “regular” classroom, is accommodated by today’s educational system in America at the expense of all the other capable students in the classroom who suffer from continual distractions and lost teaching time during the school day. Even a full-time aide who can whisk the child from the classroom when that student “loses it” cannot prevent repeated and significant learning distractions for the rest of the class. The best hope for the teacher: After many weeks have passed and a bureaucratic battery of tests on the student indicates obvious severe learning/behavioral problems, the child might be removed from the classroom. In China, the teacher with such a behavior problem would have full discretion to immediately and permanently remove that student from class – no testing, no bureaucracy, no parental approval required. The teacher in China knows what is best for the class as a whole, and that is what counts in China. This is the “Chinese way” of education philosophy. It brings to mind an old Japanese proverb which states that “the nail that protrudes, gets hammered down.” Needless to say, that approach is a 180 degree departure from the current American way which would admonish that “the protruding nail be protected at all costs.”

The author of Little Soldiers, Lenora Chu is the American mother of two young boys whose family is residing in Shanghai, China; she experienced, first hand, the highly reported, high-achieving school system in Shanghai when one of her sons attended school, there. One experience she relates in the book supports the contentions I raise in this post concerning the authority vested in China’s schoolteachers. Ms. Chu’s son was struggling with winter asthma attacks which necessitated a rescue inhaler to deal with his attacks. When teacher Chen was approached by Ms. Chu who asked where her son could keep his inhaler in the classroom, the teacher responded that the inhaler and its use in class would create unwelcome distractions for the class and thus was not allowed. When Ms. Chu asked what she and her son’s options were, the teacher informed her that she could leave the school if not satisfied. Imagine that in America! Ms. Chu realized that “going to the principal” would not change matters given the authority the system grants to classroom teachers in China. Fortunately, the boy’s asthma problem was resolved thanks to a home-administered preventative steroid inhaler.

Here are my conclusions regarding the discussion so far:

-American schools have suffered greatly from the growing lack of teacher authority in the classroom. Most of us retired folks recall our parents going into requested teacher/parent conferences ready and willing to relegate top priority to the teacher’s remarks and to their side of the story. Today, too many parents enter into discussions prepared to defend their student’s version of events despite what the teacher has to say: The “Johnny can do no wrong” syndrome is alive and well in America, but certainly not in China.

-American schools must reverse the trend and put the interests of the majority of students ahead of those individual students who require special help. I am all for funding special education classes and teachers who can help those students with severe problems, but does it make any sense to try to “include” them in regular classrooms when, by definition, they will not be able to keep pace there and will detract from the learning experience of students ready, able, and willing to learn? In that respect, the Chinese have their priorities straight.

-My family’s combined educational experiences, here, in California’s tech-savvy “Silicon Valley,” have shown that Asian and Indian students tend to display greater focus and discipline in their approach to school and education than do other students. I believe this is the by-product of cultural influences which emphasize a respect for learning and knowledge. It is an attitude formed primarily by parental and peer example and it influences students positively, especially at an early age.

-My two granddaughters are currently students in high school and junior high. They are excellent students who work hard and spend many long hours on homework assignments each week. I know that for a fact. They attend good schools which have excellent achievement records. They DO experience self-imposed and peer-imposed pressure to do well in their studies, but even their experiences likely pale in comparison to those students in Shanghai, China who face extreme pressure from home and from society to excel in school.

-I favor taking the best of both worlds which define American and Chinese education. I believe teachers in America should have much more authority in their classrooms and more respect from students, parents, and administrators. Accordingly, better pay and greater prestige for teachers should serve to attract the best and brightest to the profession. Students should come to class with a “learning attitude” which can best be nurtured at home; often in America, this is not the case.

-The Chinese system is too demanding and disciplined, overall. The fallout rate (failure rate for life, essentially) of students is unacceptable. Regrettably, the extreme discipline and enforced learning of the Chinese system can easily strangle student curiosity and creative thought, and the presence of those two key factors is the real key to an optimal educational experience for students.

I have only begun to touch upon the issues important in any discussion of students, schools, and education. So much of successful learning by students emanates not from the schools and teachers, but from parents/guardians and the home environment. Unbridled curiosity is the key catalyst for success in school. My book, Nurturing Curiosity and Success in Science, Math, and Learning explores that concept in detail. As Albert Einstein once insisted, “I have no special talents. I am only passionately curious.” And he was.
My book is not only for parents whose students are underperforming in school, but also for new and prospective parents who wish to instill a “learning attitude” in their children. And, yes, for you parents who are wondering, I write at length about the student distractions of today – namely cell phones and social media!

Isaac Newton, the most incisive mind in the history of science, reportedly uttered that sentiment about human nature. Why would he infer such negativity about his fellow humans? Newton’s scientific greatness stemmed from his ability to see well beyond human horizons. His brilliance was amply demonstrated in his great book, Philosophiae Naturalis Principia Mathematica in which he logically constructed his “system of the world,” using mathematics. The book’s title translates from Latin as Mathematical Principles of Natural Philosophy, often shortened to “the Principia” for convenience.

The Principia is the greatest scientific book ever published. Its enduring fame reflects Newton’s ground-breaking application of mathematics, including aspects of his then-fledgling calculus, to the seemingly insurmountable difficulties of explaining motion physics. An overwhelming challenge for the best mathematicians and “natural philosophers” (scientists) in the year 1684 was to demonstrate mathematically that the planets in our solar system should revolve around the sun in elliptically shaped orbits as opposed to circles or some other geometric path. The fact that they do move in elliptical paths was carefully observed by Johann Kepler and noted in his 1609 masterwork, Astronomia Nova.

In 1687, Newton’s Principia was published after three intense years of effort by the young, relatively unknown Cambridge professor of mathematics. Using mathematics and his revolutionary new concept of universal gravitation, Newton provided precise justification of Kepler’s laws of planetary motion in the Principia. In the process, he revolutionized motion physics and our understanding of how and why bodies of mass, big and small (planets, cannonballs, etc.), move the way they do. Newton did, indeed, as he stated, show us in the Principia how to calculate the motion of heavenly bodies.

In his personal relationships, Newton found dealing with people and human nature to be even more challenging than the formidable problems of motion physics. As one might suspect, Newton did not easily tolerate fools and pretenders in the fields of science and mathematics – “little smatterers in mathematicks,” he called them. Nor did he tolerate much of basic human nature and its shortcomings.

In the Year 1720, Newton Came Face-to-Face withHis Own Human Vulnerability… in the “Stock Market!”

In 1720, Newton’s own human fallibility was clearly laid bare as he invested foolishly and lost a small fortune in one of investing’s all-time market collapses. Within our own recent history, we have had suffered through the stock market crash of 1929 and the housing market bubble of 2008/2009. In these more recent “adventures,” society and government had allowed human nature and its greed propensity to over-inflate Wall Street to a ridiculous extent, so much so that a collapse was quite inevitable to any sensible person…and still it continued.

Have you ever heard of the great South Sea Bubble in England? Investing in the South Sea Trading Company – a government sponsored banking endeavor out of London – became a favorite past-time of influential Londoners in the early eighteenth century. Can you guess who found himself caught-up in the glitter of potential investment returns only to end up losing a very large sum? Yes, Isaac Newton was that individual along with thousands of others.

It was this experience that occasioned the remark about his own inability to calculate the madness of men (including himself)!

Indeed, he should have known better than to re-enter the government sponsored South Sea enterprise after initially making a tidy profit from an earlier investment in the stock. As can be seen from the graph below, Newton re-invested (with a lot!) in the South Sea offering for the second time as the bubble neared its peak and just prior to its complete collapse. Newton lost 20,000 English pounds (three million dollars in today’s valuations) when the bubble suddenly burst.

Clearly, Newton’s comment, which is the theme of this post, reflects his view that human nature is vulnerable to fits of emotion (like greed, envy, ambition) which in turn provoke foolish, illogical behaviors. When Newton looked in the mirror after his ill-advised financial misadventure, he saw staring back at him the very madness of men which he then proceeded to rail against! Knowing Newton through the many accounts of his life that I have studied, I can well imagine that his financial fiasco must have been a very tough pill for him to swallow. Many are the times in his life that Newton “railed” to vent his anger against something or someone; his comment concerning the “madness of men” is typical of his outbursts. Certainly, he could disapprove of his fellow man for fueling such an obvious investment bubble. In the end, and most painful for him, was his realization that he paid a stiff price for foolishly ignoring the bloody obvious. For anyone who has risked and lost on the market of Wall Street, the mix of feelings is well understood. Even the great Newton had his human vulnerabilities – in spades, and greed was one of them. One might suspect that Newton, the absorbed scientist, was merely naïve when it came to money matters.

That would be a very erroneous assumption. Sir Isaac Newton held the top-level government position of Master of the Mint in England, during those later years of his scientific retirement – in charge of the entire coinage of the realm!

In the year 1900, two critical questions haunted physicists, and both involved that elusive entity, light. The ultimate answers to these troublesome questions materialized during the dawn of the twentieth century and resulted in the most recent two of the four major upheavals in the history of physics. Albert Einstein was responsible for the third of those four upheavals in the form of his theory of special relativity which he published in 1905. Einstein’s revolutionary theory was his response to one of those two critical questions facing physics in the year 1900. A German scientist named Max Planck addressed the second critical question while igniting the fourth great upheaval in the history of physics. Max Planck began his Nobel Prize-winning investigation into the nature of heat/light radiation in the year 1894. His later discovery of the quantized nature of such radiation gave birth to the new realm of quantum physics which, in turn, led to a new picture of the atom and its behavior. Planck’s work directly addressed the second critical question nagging science in 1900. The aftermath of his findings ultimately changed physics and man’s view of physical reality, forever.

What were the two nagging problems in physics in 1900?

The nature of light and its behavior had long challenged the best minds in physics. For example: Is light composed of “particles,” or does it manifest itself as “waves” travelling through space? By the eighteenth century, two of science’s greatest names had voiced their opinions. Isaac Newton said that light is “particle” in nature. His brilliant French contemporary, Christian Huygens, claimed that light is comprised of “waves.”

Isaac Newton Christian Huygens

By 1865, the great Scottish physicist, James Clerk Maxwell, had deduced that light, indeed, acted as an electromagnetic wave traveling at a speed of roughly 186,000 miles per second! Maxwell’s groundbreaking establishment of an all-encompassing electromagnetic theory represents the second of the four major historical revolutions in physics of which we speak. Ironically, this second great advance in the history of physics with its theoretically established speed of light led directly to the first of the two nagging issues facing physics in 1900. To understand that dilemma, a bit of easily digestible background is in order!

Maxwell began by determining that visible light is merely a small slice of the greater electromagnetic wave frequency spectrum which, today, includes radio waves at the low frequency end and x-rays at the high frequency end. Although the speed of light (thus all electromagnetic waves) had been determined fairly accurately by experiments made by others prior to 1865, Maxwell’s ability to theoretically predict the speed of light through space using the mathematics of his new science of electrodynamics was a tribute to his supreme command of physics and mathematics. The existence of Maxwell’s purely theoretical (at that time) electromagnetic waves was verified in 1887 via laboratory experiment conducted by the German scientist, Heinrich Hertz.

The first of the two quandaries on physicist’s minds in 1900 had been brewing during the latter part of the nineteenth century as physicists struggled to define the “medium” through which Maxwell’s electromagnetic waves of light propagated across seemingly empty space. Visualize a small pebble dropped into a still pond: Its entry into the water causes waves, or ripples, to propagate circularly from the point of disturbance. These “waves” of water represent mechanical energy being propagated across the water. Light is also a wave, but it propagates through space and carries electromagnetic energy.

Here is the key question which arose from Maxwell’s work and so roiled physics: What is the nature of the medium in presumably “empty space” which supports electromagnetic wave propagation…and can we detect it? Water is the obvious medium for transmitting the mechanical energy waves created by a pebble dropped into it. Air is the medium which is necessary to propagate mechanical sound-pressure waves to our ears – no air, no sound! Yet light waves travel readily through “empty space” and vacuums!

Lacking any evidence concerning the nature of a medium suitable for electromagnetic wave propagation, physicists nevertheless came up with a name for it….the “ether,” and pressed on to learn more about its presumed reality. Clever but futile attempts were made to detect the “ether sea” through which light appears to propagate. The famous Michelson-Morley experiments of 1881 and 1887 conclusively failed to detect ether’s existence. Science was forced to conclude that there is no detectable/describable medium! Rather, the cross-coupled waves of Maxwell’s electric and magnetic fields which comprise light (and all electromagnetic waves) “condition” the empty space of a perfect vacuum in such a manner as to allow the waves to propagate through that space. In expressing the seeming lack of an identifiable transmission medium and what to do about it, the best advice to physicists seemed: “It is what it is….deal with it!”

“Dealing with it” was easier said than done, because one huge problem remained. Maxwell and his four famous “Maxwell’s equations” which form the framework for all electromagnetic phenomena calculate one and only ONE value for the speed of light – everywhere, for all observers in the universe. One single value for the speed of light would have worked for describing its propagation speed relative to an “ether sea,” but there is no detectable ether sea!

The Great “Ether Conundrum” – Addressed by Einstein’s Relativity

In the absence of an ether sea through which to measure the speed of light as derived by Maxwell, here is the problem which results, as illustrated by two distant observers, A and B, who are rapidly traveling toward each other at half the speed of light: How can a single, consistent value for the speed of light apply both to the light measured by observer A as it leaves his flashlight (pointed directly at observer B) and observer B who will measure the incoming speed of the very same light beam as he receives it? Maxwell’s equations imply that each observer must measure the same beam of light at 186,000 miles per second, measured with respect to themselves and their surroundings – no matter what the relative speed between the two observers. This made no sense and represented a very big problem for physicists!

The Solution and Third Great Revolution in Physics: Einstein’s Relativity Theories

As already mentioned, the solution to this “ether dilemma” involving the speed of light was provided by Albert Einstein in his 1905 theory of special relativity – the third great revolution in physics. Special relativity completely revamped the widely accepted but untenable notions of absolute space and absolute time – holdovers from Newtonian physics – and time and space are the underpinnings of any notion/definition of “speed.” Einstein showed that a strange universe of slowing clocks and shrinking yardsticks is required to accommodate the constant speed of light for all observers regardless of their relative motion to each other. Einstein declared the constant speed of light for all observers to be a new, inviolable law of physics. Furthermore, he proved that nothing can travel faster than the speed of light.

The constant speed of light for all observers coupled with Einstein’s insistence that there is no way to measure one’s position or speed/velocity through empty space are the two notions which anchor special relativity and all its startling ramifications.

The Year is 1900: Enter Max Planck and Quantum Physics –The Fourth Great Revolution in Physics

The second nagging question facing the physics community in 1900 involved the spectral nature of radiation emanating from a so-called black-body radiator as it is heated to higher and higher temperatures. Objects that are increasingly made hotter emanate light whose colors change from predominately red to orange to white to a bluish color as the temperature rises. A big problem in 1900 was this: There is little experimental evidence indicating large levels of ultraviolet radiation produced at high temperatures – a situation completely contrary to the theoretical predictions of physics based on our scientific knowledge in the year 1900. Physics at that time predicted a so-called “ultraviolet catastrophe” at high temperatures generating huge levels of ultraviolet radiation – enough to damage the eyes with any significant exposure. The fact that there was no evidence of such levels of ultraviolet radiation was, in itself, a catastrophe for physics because it called into serious question our knowledge and assumptions of the atomic/molecular realm.

The German physicist, Max Planck, began tackling the so-called “ultraviolet catastrophe” disconnect as early as 1894. Using the experimental data available to him, Planck attempted to discern a new theory of spectral radiation for heated bodies which would match the observed results. Planck worked diligently on the problem but could not find a solution by working along conventional lines.

Finally, he explored an extremely radical approach – a technique which reflected his desperation. The resulting new theory matched the empirical results perfectly!

When Planck had completed formulation of his new theory in 1900, he called his son into his study and stated that he had just made a discovery which would change science forever – a rather startling proclamation for a conservative, methodical scientist. Planck’s new theory ultimately proved as revolutionary to physics as was Einstein’s theory of relativity which would come a mere five years later.

Max Planck declared that the radiation energy emanating from heated bodies is not continuous in nature; that is, the energy radiates in “bundles” which he referred to as “quanta.” Furthermore, Planck formulated the precise numerical values of these bundles through his famous equation which states:

E = h times Frequency

where “h” is his newly-declared “Planck’s constant” and “Frequency” is the spectral frequency of the radiation being considered. Here is a helpful analogy: The radiation energy from heated bodies was always considered to be continuous – like water flowing through a garden hose. Planck’s new assertion maintained that radiation comes in bundles whose “size” is proportional to the frequency of radiation being considered. Visualize water emanating from a garden hose in distinct bursts rather than a continuous flow! Planck’s new theory of the energy “quanta” was the only way he saw fit to resolve the existing dilemma between theory and experiment.

The following chart reveals the empirical spectral nature of black-body radiation at different temperatures. Included is a curve which illustrates the “ultraviolet catastrophe” at 5000 degrees Kelvin predicted by (1900) classical physics. The catastrophe is represented by off-the-chart values of radiation in the “UV” range of short wavelength (high frequency).

This chart plots radiated energy (vertical axis) versus radiation wavelength (horizontal axis) plotted for each of three temperatures in degrees K (degrees Kelvin). The wavelength of radiation is inversely proportional to the frequency of radiation. Higher frequency ultraviolet radiation (beyond the purple side of the visible spectrum) is thus portrayed at the left side of the graph (shorter wavelengths).

Note the part of the radiation spectrum which consists of frequencies in the visible light range. The purple curve for 5000 degrees Kelvin has a peak radiation “value” in the middle of the visible spectrum and proceeds to zero at higher frequencies (shorter wavelengths). This experimental purple curve is consistent with Planck’s new theory and is drastically different from the black curve on the plot which shows the predicted radiation at 5000 degrees Kelvin using the scientific theories in place prior to 1900 and Planck’s revolutionary findings. Clearly, the high frequency (short wavelength) portion of that curve heads toward infinite radiation energy in the ultraviolet range – a non-plausible possibility. Planck’s simple but revolutionary new radiation law expressed by E = h times Frequency served to perfectly match theory with experiment.

Why Max Planck Won the 1918 Nobel Prizein Physics for His Discovery of the Energy Quanta

One might be tempted to ask why the work of Max Planck is rated so highly relative to Einstein’s theories of relativity which restructured no less than all of our assumptions regarding space and time! Here is the reason in a nutshell: Planck’s discovery led quickly to the subsequent work of Neils Bohr, Rutherford, De Broglie, Schrodinger, Pauli, Heisenberg, Dirac, and others who followed the clues inherent in Planck’s most unusual discovery and built the superstructure of atomic physics as we know it today. Our knowledge of the atom and its constituent particles stems directly from that subsequent work which was born of Planck and his discovery. The puzzling non-presence of the “ultraviolet catastrophe” predicted by pre-1900 physics was duly answered by the ultimate disclosure that the atom itself radiates in discrete manners thus preventing the high ultraviolet content of heated body radiation as predicted by the old, classical theories of physics.

Albert Einstein in 1905: The Photoelectric Effect –Light and its Particle Nature

Published in the same 1905 volume of the German scientific journal, Annalen Der Physik, as Einstein’s revolutionary theory of special relativity, was his paper on the photoelectric effect. In that paper, Einstein described light’s seeming particle behavior. Electrons were knocked free of their atoms in metal targets by bombarding the targets with light in the form of energy bundles called “photons.” These photons were determined by Einstein to represent light energy at its most basic level – as discrete bundles of light energy. The governing effect which proved revolutionary was the fact that the intensity of light (the number of photons) impinging on the metal target was not the determining factor in their ability to knock electrons free of the target: The frequency of the light source was the governing factor. Increasing the intensity of light had no effect on the liberation of electrons from their metal atoms: The frequency of the light source had a direct and obvious effect. Einstein proved that these photons, these bundles of light energy which acted like bullets for displacing electrons from their metal targets, have discrete energies whose values depend only on the frequency of the light itself. The higher the frequency of the light, the greater is the energy of the photons emitted. As with Planck’s characterization of heat radiation from heated bodies, photon energies involve Planck’s constant and frequency. Einstein’s findings went beyond the quanta energy conceptualizations of Planck by establishing the physical reality of light photons. Planck interpreted his findings on energy quanta as atomic reactions to stimulation as opposed to discrete realities. Einstein’s findings earned him the 1921 Nobel Prize in physics for his paper on the photoelectric effect….and not for his work on relativity!

Deja Vu All Over Again: Is Light a Particle or a Wave?

Along with Planck, Einstein is considered to be “the father of quantum physics.” The subsequent development by others of quantum mechanics (the methods of dealing with quantum physics) left Einstein sharply skeptical. For one, quantum physics and its principle of particle/wave duality dictates that light behaves both as particle and wave – depending on the experiment conducted. That, in itself, would trouble a physicist like Einstein for whom deterministic (cause and effect) physics was paramount, but there were other, startling ramifications of quantum mechanics which repulsed Einstein. The notion that events in the sub-atomic world could be statistical in nature rather than cause-and-effect left Einstein cold. “God does not play dice with the universe,” was Einstein’s opinion. Others, like the father of atomic theory, Neils Bohr, believed the evidence undeniable that nature is governed at some level by chance.

In one of the great ironies of physics, Einstein, one of the two fathers of quantum physics, felt compelled to abandon his brain-child because of philosophical/scientific conflicts within his own psyche. He never completely came to terms with the new science of quantum physics – a situation which left him somewhat outside the greater mainstream of physics in his later years.

Like Einstein’s relativity theories, quantum physics has stood the test of time. Quantum mechanics works, and no experiments have ever been conducted to prove the method wrong. Despite the truly mysterious realm of the energy quanta and quantum physics, the science works beautifully. Perhaps Einstein was right: Quantum mechanics, as currently formulated, may work just fine, but it is not the final, complete picture of the sub-atomic world. No one could appreciate that possibility in the pursuit of physics more than Einstein. After all, it was his general theory of relativity in 1916 which replaced Isaac Newton’s long-held and supremely useful force-at-a-distance theory of gravity with the more complete and definitive concept of four-dimensional, curved space-time.

By the way, and in conclusion, it is Newton’s mathematics-based science of dynamics (the science of force and motion) that defines the very first major upheaval in the history of physics – as recorded in his masterwork book from 1687, the Principia – the greatest scientific book ever written. Stay tuned.

Yes, it is back-to-school time for many of the world’s youngsters. In America, late August and early September is when students return to school to meet new teachers who will be entrusted by parents to help educate their children.

Have you, as parents, guardians, or mentors nurtured your student’s curiosity this summer? My book on education, learning, and mentoring suggests that successful learning and top student performance stem from a healthy curiosity – the desire to know and understand the world around us. Such a “learning attitude” (or lack thereof) is influenced primarily by the home environment and the adults at home – not by the students’ school and teachers. Equipped with a good “learning attitude” acquired in the home, students prosper at school; without a proper attitude, many disinterested youngsters flounder in class while being easily distracted by social media and the associated electronic connectedness so prevalent today.

Sadly, many of these children will, in the course of their schooling, waste the most precious opportunity that society will ever offer them – a good education and a pathway to lifelong learning. It need not be that way, however.

My book is a hands-on, how-to manual for parenting/mentoring with the end goal of insuring school success for students – especially in science and mathematics.

Nurturing Curiosity and Success in Science, Math, and Learning, is available from Amazon for $14.95. This link will take you directly to Amazon and the book.

Aside from love, one of the great emotions humans can experience is the thrill of discovery and achievement – being the first to reveal more of nature’s immutable laws governing the cosmos or doing something no one else has been able to do. Some aspects of life inevitably go together – a coupling of cause-and-effect, if you will. Sometimes, we simply cannot have one thing without another. The claim that “there is a price to be paid for everything” seems a truism which ably illustrates that contention of coupled cause-and-effect. In that vein, man’s finest intellectual achievements or physical accomplishments materialize only after significant vested effort is expended. Our personal life experiences leave no doubt that hard work is a necessary, though not sufficient, prerequisite for great success…in any venue. We understand that. Not so obvious is the other price often associated with intellectual achievement and intellectual property, a price which is extracted after the fact – the tedious, ongoing, and costly effort required to establish and maintain the legal rights to the intellectual property behind any significant achievement.

I call this second price to be paid for success “the indigestion of success” which is often so severe as to result literally in ulcers if not merely pervasive, never-ending discontent.

The “indigestion of success” begins with proving one’s priority of invention while establishing patent rights, and it continues seemingly forever while vigilantly protecting those rights against usurpers. The motivation to defend one’s intellectual property is typically financial, but, understandably, the battle becomes distinctly a matter of personal principle as we will see…and the consequences can be tragic. It is difficult to overstate the high price – both financially and emotionally – of defending intellectual property and priority, yet this surcharge on success is inevitably demanded of inventors, engineers, scientists, and entrepreneurs. The list of such examples is varied and fascinating, stretching far back in recorded history. Gaileo Galilei, Isaac Newton, and Michael Faraday, three of the greatest physicists of all time were each affected by priority controversies during their careers – especially Newton, as we shall see. In the realms of engineering and business, Thomas Edison, Howard Armstrong (radio’s greatest inventor/engineer), Robert Noyce (of integrated circuit fame), and Steve Jobs of Apple Computer were all enveloped by priority controversies and patent battles. Even the Wright brothers, the well-documented founders of modern aviation paid a stiff price defending their marvelous invention, the controllable “flying machine.”

The Wright Brothers: Hard Work, Triumph, then Disillusionment

I just finished reading David McCullough’s new book, The Wright Brothers, which relates the incredible story of the two brothers from Dayton, Ohio – bicycle mechanics/salesmen who created the first true “flying machine”…in their spare time! McCullough is a consummate teller of true stories, but the story of these two men tests the line separating fact from fiction because their stunning success seemed so improbable. The truth is, the Wright Brothers “invented” and successfully flew the first full-sized, self-powered, controllable airplane – a staggering accomplishment for two young men with no formal technical credentials. Their ultimate success was rooted in a fascination at the prospect of manned flight coupled with a single-minded, driven determination to do whatever it takes to accomplish their dream of flying. The two brothers constitute the very best examples of self-made men… engineers and flyers, in their case. Their accomplishments are so thoroughly documented as to seem unassailable and safe from thieves who would steal in the courts of patent law, yet it was not quite that simple. It never is. Author McCullough paints a clear picture on his pages of just how technically challenging their task actually was. What also emerges is the sad turn of events their triumph became once the airplane was designed, tested, documented, and patented. Wilbur Wright, the brilliant engineering mind for whom no technical challenge seemed too large, died early in 1912 at the young age of forty-five years. The official cause of death was typhoid fever, but it seems Wilbur’s spirit was dying for quite some time before his body expired. In May of 1910, the brothers, who did all their own flying from the project’s onset in 1900, went up together in their Wright Flyer for the very first time – some seven years after Orville’s first flight at Kitty Hawk. Their disciplined methodology throughout the project dictated that, should there be an accident, at least one of them should survive to carry on the work. Their flight together that day seemed their tacit acknowledgement that they had completed their life’s dream; all that remained was to form and grow a profitable company which would carry on their work and insure a comfortable livelihood for the brothers and their immediate relatives.

By 1912, two years had passed since Wilbur Wright had last done what he truly loved to do: Piloting the Wright Flyer while perfecting its design. His weeks and months the past two years were spent on business trips to New York and Washington and in courtrooms defending the patent portfolio he and Orville had assembled as the backbone of their new Wright Company… for the manufacture of airplanes. In author McCullough’s account, Orville took note of Wilbur’s restless discontent with the tedium and exasperations of establishing their company, noting that after a day spent in offices dealing with business and patent matters, Wilbur would “come home white.”

Wilbur, himself, wrote of the patent entanglements: “When we think of what we might have accomplished if we had been able to devote this time to experiments, we feel very sad, but it is always easier to deal with things than with men, and no one can direct his life entirely as he would choose.”

Within several years of Wilbur’s death, Orville Wright had sold the Wright Company to others, preferring a peaceful, retiring life to one spent constantly battling corporate demons and those who would usurp the brothers’ past and future accomplishments. His mission for the remainder of his long life: To represent his brother while defending the less materialistic aspects of the Wright brothers’ legacy. I believe I would have done precisely the same, were I in his shoes. Other notable, historical figures in similar circumstances made sadly different decisions when faced with the indigestion of success and the never-ending need to protect intellectual property. The two examples that follow vividly illustrate just how bad these matters of priority and intellectual property can become, especially for the most-principled of participants.

Edwin Howard Armstrong: Radio’s Greatest Inventor/Engineer and Tragic Victim of His Own Success and the Patent System

For radio and electrical engineers who know the history, Edwin Howard Armstrong is the tragic hero of early “wireless” and a victim of the radio empire which he helped to create. Howard Armstrong was the quintessential radio engineer’s engineer – bright, motivated, creative…and stubbornly persistent. He exuded personal integrity. The very qualities which made him the greatest inventor/engineer in the history of radio, led to his downfall and suicide in 1954. Howard Armstrong surfaced in 1912 as a senior electrical engineering major at Columbia University with an obsessive interest in the infant science of “wireless” radio. He was a fine student with a probing, independent mind that suffered no fools. In 1912, while living at home in nearby Yonkers, New York, and commuting daily to Columbia on an Indian-brand motorcycle, he invented a way to greatly increase signal amplification using a single De Forest Audion vacuum tube by feeding part of the tube’s marginally amplified output back to the input of the device where it was amplified over and over again. This technique is now known in the trade as “regeneration,” or positive feedback. Along the way, young Armstrong had made great strides in understanding the technology behind Lee De Forest’s recent invention of the Audion tube, insights far beyond those De Forest himself had offered. While tinkering with the idea of signal regeneration in his bedroom laboratory early on the morning of September 22, 1912, he achieved much greater signal amplification from the Audion than was possible without using regeneration. The entire household was abruptly awakened by young Armstrong’s unrestrained excitement over his discovery, and an important discovery it was for the infant science of “wireless radio.” Regeneration was patented by Armstrong in 1913/14 and was used, under license from him, in countless radios during the early years when radio sets with more than one tube were very expensive to produce, due to the high cost of tubes.

Armstrong’s 1914 patent on the regenerative receiving circuit – one of the foundations of early wireless radio and a gateway to efficient tube-based radio transmitters, as well.Armstrong’s historic, handwritten chronological account of inventing the regenerative circuit – page one of six; likely written around 1920 to serve as evidence in the litigation with De Forest over Armstrong’s regeneration patent. Note the Sept. 22, 1912 date of his triumph (near the bottom).

In 1914, Lee De Forest stepped forward to challenge Armstrong in court over Armstrong’s patent, claiming that he, De Forest, was the legitimate inventor of regeneration. The litigation in the court system over regeneration went back and forth, lasting twenty years and finally ending up in the United States Supreme Court. Shockingly, De Forest was handed the final decision by the court, but the substantial body of radio engineers across the nation in 1934, who were well aware of the “radio art” and its history, were not buying De Forest’s claim. They fully supported Armstrong as the legitimate inventor – the same view held today. The twenty-year patent litigation battle over regeneration was the longest in U.S. patent court history. Unfortunately, that was only the beginning of Armstrong’s troubles with the patent courts and those who would take advantage of his work.

The Tragedy of Edwin Howard Armstrong

Howard Armstrong was one of the last, great, lone-inventor/engineers. He was long affiliated with his alma-mater, Columbia University, and had extensive business/patent dealings with giant corporations, such as RCA and Philco, which drew their life-blood from his inventions and the industry which he helped to create. By licensing his many important patents to these corporations, Armstrong became a very wealthy man. At one time, he was the largest stockholder in the giant RCA Corporation. Despite such wide-spread affiliations, he was, by temperament, an independent thinker in the lone-inventor mold. As radio entered the late nineteen thirties, men-of-action like Armstrong were becoming obsolete, increasingly overrun by corporate bureaucracies and their in-house armies of engineers. Radio was now out of the hands of the lone-inventor, becoming the exclusive domain of the moneyed corporations with influence at the FCC (Federal Communications Commission) in Washington. Armstrong increasingly found himself defending his legitimate patent rights against large corporations which were treading on those rights, battling their great financial resources and their legions of corporate lawyers. As he continued to lose rightful patent royalties to corporate violations of his patents, he stubbornly fought back fueled by his personal principles of fair play, all the while dissipating his once-great financial security to fund the necessary lawyer’s fees. Armstrong was a man of principled integrity; he could have capitulated, retreated, retired comfortably, and lived out his life, but he chose to fight.

Ultimately, those ceaseless legal battles wore him down, bankrupted him, and destroyed his long marriage. On May 5, 1954, he stepped from his New York apartment window to his death thirteen stories below. In an ironic sense, he fell victim to the industry and the changing times he helped to create. He also was victimized by the very qualities which made him great: Intellectual independence, principled integrity, and the stubborn will to persevere. There are many lessons to be learned from Howard Armstrong’s life-story. The lone crusader was crushed by the corporate “Goliaths” he helped create. Final postscript: After Armstrong’s death, his estranged wife, Marion, took up her husband’s ongoing patent battles with the Goliaths of the radio industry. She eventually prevailed in every single case!

History’s most ardent defender of his intellectual property also happened to be the greatest scientist/mathematician of all time, Sir Isaac Newton. As vindictive as he was brilliant, Newton waged one of history’s most vicious priority battles with Gottfried Leibniz over credit for the development of the calculus, that ubiquitous, indispensable mathematical tool of the engineer and scientist. Newton formulated its fundamentals in 1665/66, the famous “miracle year” spent at his mother’s homestead in isolation from the great plague which swept through England at the time. Newton’s peerless scientific self-discipline tended to completely desert him when challenged by others on matters of intellectual priority which he felt belonged to him. Leibniz and Robert Hooke were two men who famously felt the full force of Newton’s rage in such matters. For Newton and his circumstances, there was no real money at stake – only prestige and ego, and Newton’s ego was well-developed… and sensitive. Today, both Newton and Leibniz are credited with independently developing the calculus – essentially true, although it appears certain that Leibniz had unauthorized access to some of Newton’s early personal papers on the subject. In that sense, Newton is regarded as the “primary” developer of calculus. Leibniz never quite recovered from the savage and telling effects of Newton’s vindictiveness which was well publicized in scientific circles and which reduced the great Newton to unprincipled deceits in his efforts to discredit his rival. In Newton’s mind, much more was at stake than mere money: For him, personal satisfaction and the ego-satisfying prospect of scientific immortality were far more important motivators. In his defense, one could argue that, for Newton, the long-term stakes riding on his efforts to receive due credit for his brilliance were much higher than most. Nevertheless, when all was said and done, Newton’s personal reputation suffered significantly even if his scientific reputation remained unsullied over the dispute with Leibniz.

What Would You Do?

If you were ever in the position of enjoying a significant personal success that had already conferred substantial wealth upon you, yet huge wealth beckons you or whoever else takes the enterprise still further – what would you do? Like Orville did, I would have heeded Bishop Milton Wright’s early admonition to his children (paraphrased here) that greed is bad and leads to grief; be content with sufficient money to sustain a comfortable life and require nothing more beyond that than the normal pleasures of life and living. The Bishop also warned against temper and ego. The Bishop was a very wise man; the brothers received some very informed guidance.

“If I were giving a young man advice as to how he might succeed in life, I would say to him, pick out a good father and mother, and begin life in Ohio.”

I had a very good day recently. I bought a beautiful $400 book for $20 in Ventura, California! It also happens to be a very important book – literally, a foundation work for today’s Internet and our computer-based technological age . The book is titled: Cybernetics. While traveling south three weeks ago to the annual Santa Clarita Cowboy Music Festival – an annual event for us (see last week’s post) – Linda and I stopped in downtown Ventura, California – also an annual ritual. As always, we had lunch at our favorite hole-in-the-wall Italian restaurant and browsed for a bit at one of our favorite used bookstores in town, The Calico Cat.

Often we find a book or two in this little shop, and sometimes, we do not. After perusing various sections for close to an hour with no luck, I moved to the science/math section. As I ran my eyes along the shelves, I recognized many of the books they held. My scanning gaze froze as I came upon a pristine little book titled Cybernetics: or Control and Communication of the Animal and the Machine, written by Norbert Wiener. Wiener was a mathematics prodigy in his youth who enjoyed a long and distinguished career as a professor of mathematics at M.I.T., the Massachusetts Institute of Technology.

The book’s title reflects its ground-breaking categorization of the messaging and control systems inherent in the two closely-related realms of computer control and the human/animal brain/body connection. Cybernetics appeared at precisely the same time as the first, large-scale electronic computers, and this little book was instrumental in determining the future path of computing and control (robotics).

I immediately recognized the title and author as very important, but could this be the 1948 first edition – a book I knew to be of considerable value? I excitedly pulled it from the shelf and opened to the verso of the title page which stated “Second printing. November, 1948.” I was holding the second printing of the first U.S. edition printed by John Wiley and Sons, Inc. There was a companion edition of the text published in Paris by Hermann et CIE, also in 1948.

I became very excited and called Linda over to show her the book and explain, “I believe this book is worth several hundred dollars in the book trade: It is a very famous, seminal work in communication and control engineering. For a copy in the like-new condition of this one, $400 is easily a realistic value on the market. The apparent penciled price for this pristine copy with an almost perfect dust jacket: $25! The store’s owner called my attention to the fact I misread the price which was actually only $20. With no hint of hesitation, I coolly announced, “I’ll take it!” The book’s original price, still on the dust-jacket: $3! Understandably, the store owners had no clue as to the book’s engineering/mathematical significance to today’s Internet and computer technology.

For a retired electrical engineer, like myself, finding this little book in such perfect condition at such a price is akin to tripping over a diamond protruding from the footpath. My many years in Silicon Valley spent designing computer disk drives all ultimately stemmed from a very few foundational works (books and technical papers) such as this one. Summoning my engineering background, I can read and understand the material in this book –while difficult, it does not require a PhD in Mathematics. That is the beauty of a foundational technical work such as this – profound, yet accessible to most engineers and scientists – given some effort.

The “Other” Book

There exists another similarly concise book whose pedigree exceeds even that of Cybernetics. That book was authored by Claude Shannon at the Bell Laboratories and titled The Mathematical Theory of Communication. Interestingly, the book was published in 1949 after being first introduced as a technical paper in the Bell Systems Journal of 1948 – the same year that Cybernetics was published.

Dare I press my luck and hope to find a similar bargain for Mr. Shannon’s book? Not likely to happen, but it would be the perfect complement to Wiener’s little volume.

Shannon’s book elegantly achieves the unenviable task of defining “information” in mathematical terms and in a manner which lends itself to quantifying the maximum flow of information possible over a given communication channel such as the Internet or the radio/television airwaves, to cite two of many possible real-life applications. Reading and decoding the magnetically recorded binary bits of data (1’s and 0’s) stored on computer disk drives occurs in the “read channel” of the drive electronics, as we engineers in the industry referred to it. All such applications concerning “communications” succumb to the mathematics presented by Claude Shannon in this little volume containing a mere 117 pages! Shannon’s methods are equally applicable to yesterday’s analog channels (radio transmissions, for example) and today’s pervasive digital implementations (computers, the Internet, et all).

The next time we are passing through Ventura, I will keep a sharp eye out for this second book and any other bargains like Cybernetics! Good fortune usually takes luck, but when good luck comes knocking, one needs to recognize the sound!